In the previous lesson, the discount rate k was defined by a vague phrase: "risk-free rate plus a risk premium." But what exactly is risk? How is it measured? And why can two risky assets, combined, form a portfolio that is safer than either one on its own?This lesson builds the answer piece by piece—the insight that earned Harry Markowitz a Nobel Prize: risk is measured by the dispersion of returns, combined through correlation, and reduced through diversification—the only free lunch in finance. By the end, you will understand why "don't put all your eggs in one basket" went from a grandmother's proverb to a mathematical theorem.1. Risk Is Dispersion: Expected Return and DeviationsIn finance, risk is not the "chance of something bad happening"—it is uncertainty about the outcome, in both directions. An asset whose return can be −20% or +40% is riskier than one that fluctuates between +8% and +12%, even if both have the same average.The starting point is expected return: the average of the possible returns, weighted by the probability of each scenario.E[R] = Σ (scenario probability × return in the scenario)1.1 Worked ExampleCia. Ciclotur's stock has three scenarios for the coming year:Recession (probability 30%): return of −10%. Normal conditions (40%): +10%. Expansion (30%): +30%.E[R] = 0,30 × (−10%) + 0,40 × 10% + 0,30 × 30% = −3% + 4% + 9% = 10% per yearA stock has a 50% probability of returning 20% and a 50% probability of returning 0%. Its expected return is...Answer: 10% — E[R] = 0,5 × 20% + 0,5 × 0% = 10%. Notice that the "expected" return may never actually occur (the possible outcomes are 0% or 20%)—it is the center of gravity of the possibilities, not a promise.2. Volatility: Putting a Number on NervousnessTwo assets can have the same E[R] and completely different temperaments. The measure of that restlessness is variance—the weighted average of the squared deviations from the expected return—and, more importantly, its square root: standard deviation, which the market calls volatility (σ), expressed in the same units as the return (% per year).Continuing with Ciclotur (E[R] = 10%):Deviations: recession: −10 − 10 = −20 p.p.; normal conditions: 0; expansion: +20 p.p.Variance = 0,30 × (−20)² + 0,40 × 0² + 0,30 × (20)² = 120 + 0 + 120 = 240Volatility = √240 ≈ 15,5% per yearA rule of thumb: Ciclotur's returns tend to spread across a range of ±15,5 p.p. around the expected 10%. To calibrate your intuition against the real world: an LFT has near-zero volatility; the Ibovespa has historically been around 20–30% per year; an individual stock is often much higher. Higher σ = higher risk—that is the yardstick.Two funds delivered the same average return of 12% per year over the past decade. Fund Alfa had a standard deviation of 5% per year; Beta's was 25% per year. It is correct to say that...Answer: Beta was riskier because its returns were much more dispersed around the average—same average, opposite temperaments: anyone who had to withdraw money in a bad year for Beta felt the difference. Risk is dispersion, not the average.3. Markowitz's Magic: 1 + 1 Can Be Less Than 2Now comes the lesson's conceptual leap. The naive intuition is that a portfolio's risk would be the average of the assets' risks. Wrong—and modern portfolio theory grew out of that mistake.3.1 The Beach ExperimentImagine two companies in the same coastal town:The Solmar Ice Cream Shop: it makes profits during scorching summers and struggles in rainy years. E[R] = 12%, σ = 20%.The Torrente Umbrellas: the perfect mirror image—it profits in the rain and struggles in the sun. E[R] = 12%, σ = 20%.Each one on its own is a roller coaster. But what about a portfolio split 50% into each? In EVERY scenario—sun or rain—whatever one loses, the other gains. The portfolio's return is locked at 12%, with zero volatility. Two risky assets have formed a risk-free portfolio.3.2 The Secret Ingredient: CorrelationWhat made the magic possible was the correlation coefficient (ρ) between the returns—a number between −1 and +1 that measures how closely two assets move together:ρ = +1: perfectly synchronized. Diversifying between them does not reduce risk at all—the portfolio is simply the average of the risks.ρ = 0: independent dancers. Diversification already reduces risk: in our pair (two assets with σ = 20%, split evenly), portfolio volatility falls to √(0,25×400 + 0,25×400) ≈ 14,1%—below the 20% of either asset.ρ = −1: mirror-image movement, as with Solmar × Torrente. With the right weights, risk can be eliminated completely: σ = 0.The general rule that matters here: whenever ρ < 1, portfolio risk is LOWER than the weighted average of the individual risks. Since virtually no pair of real-world assets has perfect correlation, diversification almost always reduces risk without sacrificing expected return—that is the free lunch.And a grown-up warning: correlations are not fixed. During severe crises, correlations between risky assets tend to rise ("when markets fall, everything falls together")—diversification offers less protection precisely when it is needed most. Genuine protection against systemic panic requires assets of a different nature (government bonds, dollars)—not simply more different stocks.Two assets have a correlation of −1 between their returns. By combining them in a portfolio with the appropriate weights, it is possible to...Answer: Eliminate portfolio risk completely—with ρ = −1, the movements cancel each other out exactly, and there is a combination of weights with σ = 0. This is the theoretical limiting case that reveals the mechanism: joint risk depends on HOW the assets move relative to each other, not just on how much each one fluctuates.4. The Limits of the Free Lunch: Systematic × Unsystematic RiskIf diversification reduces risk, why not diversify forever and eliminate it entirely? Because the total risk of any stock has two layers with different characteristics:Unsystematic risk (specific or diversifiable): events affecting the company or sector—the factory strike, accounting fraud, a competitor that steals market share, a frost damaging the harvest. Because these are events independent across companies, the misfortunes of some are offset by the fortunes of others in a broad portfolio: this risk disappears through diversification. Classic studies show that a portfolio holding somewhere between 15 and 30 stocks spread across sectors already eliminates most of it.Systematic risk (market-wide, non-diversifiable): events that affect all companies at the same time—the Copom raising interest rates, a recession, a currency crisis, a pandemic. It does not matter how many stocks you own: when the tide goes out, all boats go down. This risk cannot be diversified away.The logical consequence, which sets up the next lesson: if specific risk can be eliminated for free, the market does not pay a premium for bearing it. The only risk that earns compensation is systematic risk—and measuring each stock's exposure to it is the job of a number called beta, the star of Lesson 12.A prolonged strike shuts down the factories of a single company, sending its stock price lower. For a diversified investor, this event represents... riskAnswer: Unsystematic risk, which portfolio diversification dilutes—the event is specific to that company; in a portfolio of dozens of securities, its impact is small and tends to be offset. A general rise in interest rates, by contrast, would be systematic: it would affect the entire portfolio.Even while holding a portfolio with hundreds of stocks from every sector, an investor remains exposed to... riskAnswer: Systematic risk, linked to factors that affect the market as a whole—interest rates, recessions, crises. Diversification sweeps away specific risk, but market risk is the irreducible floor: it is the only risk the market rewards with a premium.“Diversification is the only free lunch in finance.”- Attributed to Harry Markowitz, 1990 Nobel Prize winner5. Bringing It All TogetherThe essentials of the lesson:Expected return is the weighted average of the scenarios; risk is dispersion, measured by variance and standard deviation (volatility).Portfolio risk depends on how assets move together: their correlation (ρ), between −1 and +1. When ρ < 1, portfolio risk falls below the average of the individual risks—and with ρ = −1, it can reach zero.Diversification reduces risk without reducing expected return—but correlations rise during crises, and the free lunch gets smaller when the restaurant catches fire.Total risk is divided into unsystematic (specific, eliminated with ~15–30 diversified stocks) and systematic (market-wide, irreducible). Only systematic risk is rewarded.Next lesson: the grand finale of equity investing, portfolio theory and the CAPM. The efficient frontier, beta as a measure of the risk that matters—and, finally, the formula that turns "risk-free rate + premium" into the k that was missing from Gordon.Exercise for Classroom DiscussionThe commander's portfolio. Let's think it through together.Túlio is an aircraft commander for a large Brazilian airline, where he has worked for 15 years. Proud of the company, he has built his wealth as follows: 70% in shares of the airline itself (some purchased, some received through compensation programs), 20% in shares of two other tourism companies (a hotel chain and a travel operator), and 10% in a checking account. "I invest in what I know," he says.Discuss with your group:(a) Separate the risks in Túlio's portfolio into the two layers from the lesson: which parts represent specific risk and which represent systematic risk? Is his portfolio diversified?(b) There is a huge asset outside Túlio's financial portfolio: his human capital—the stream of future salaries. What is the correlation between this "asset" and the portfolio he has built? What happens to EVERYTHING at the same time in a scenario of surging oil prices or a pandemic?(c) Is "I invest in what I know" a good diversification principle? Where does familiarity help, and where does it become a trap?(d) Propose a portfolio overhaul using the concepts from the lesson: what types of assets have low (or negative) correlation with aviation and tourism risk? Think about sectors, fixed income (Lessons 3–8), and the role of the 10% in the checking account.(e) Even after a perfect overhaul, what risk will Túlio continue to bear? Is there any way to diversify against it?Instructor guidance: (a) The portfolio has 90% concentrated in a single risk cluster (aviation-tourism): the sector-specific risk is far from diluted — strikes, accidents, oil prices and exchange rates affect everything at once; extremely high internal correlations make 3 assets function as 1. (b) Strongly positive correlation: Túlio's salary AND his portfolio depend on the same industry — in a sector shock, he could lose both his job and his wealth simultaneously (cite real cases of employees heavily concentrated in their own employer's stock); human capital must be included in the risk map. (c) Knowledge helps you ASSESS (Lesson 10), but it does not protect against concentration — familiarity breeds overconfidence and loyalty bias; knowing the industry does not remove the correlation. (d) Defensible answers: diversify broadly across sectors with little connection to tourism (utilities, water and sanitation, banks, exporters — a fine point: oil companies rise when an airline's costs soar, providing a partial natural hedge), use a broad index ETF to eliminate specific risk in one stroke, and establish a fixed-income anchor (LFT/IPCA+) as a reserve and foundation — the 10% in the checking account earns nothing and should at least be moved into an LFT (Lesson 5). (e) Systematic risk: a global recession, interest rates and crises will also drag down the rebalanced portfolio — equities cannot be diversified against it; the only protection is allocation across ASSET CLASSES (fixed income, currencies) and the premium the market pays for bearing it, the subject of Lesson 12.Exercises1) (Original question - CESGRANRIO style) A stock has the following scenarios for the coming year: a −5% return with a 20% probability; a 10% return with a 50% probability; and a 20% return with a 30% probability. The expected return on this stock is:A) 10.0%.B) 8.3%.C) 12.5%.D) 6.0%.E) 25.0%.2) (Original question - CESGRANRIO style) An asset has two equally likely return scenarios: 4% or 16%. The expected return and standard deviation of this asset are, respectively:A) 10% and 6%.B) 10% and 36%.C) 12% and 6%.D) 10% and 12%.E) 8% and 4%.3) (Original question - CESGRANRIO style) The correlation coefficient between the returns of two financial assets can take values:A) between −1 and +1, indicating anything from perfectly opposite movements to perfectly synchronized movements.B) between 0 and +1, because negative correlations do not exist in finance.C) between 0% and 100%, like any rate.D) only −1, 0 or +1.E) any real number, without limits.4) (Original question - FGV style) The risk-reduction benefit provided by diversification between two assets is greatest when the correlation between their returns is equal to:A) −1.B) +1.C) 0.D) +0.5.E) any value, because correlation does not affect diversification.5) (Original question - FGV style) Examples of systematic (non-diversifiable) risk and unsystematic (diversifiable) risk, respectively, are:A) a generalized increase in the benchmark interest rate and an accounting fraud uncovered at a specific company.B) a strike at a company's factory and a global recession.C) a company losing a patent and another company recalling a product.D) a global pandemic and a currency crisis.E) a retailer's profits declining and a competitor's warehouse catching fire.6) (Original question - CESGRANRIO style) Two assets each have a standard deviation of 20% per year. Combined in a portfolio with 50% invested in each, and with a correlation between them BELOW +1, the portfolio's standard deviation will be:A) below 20%.B) exactly 20%.C) above 20%.D) equal to 40%.E) equal to zero, regardless of the correlation.7) (Original question - CEBRASPE style, judge the statement) Judge the following statement: "By distributing their resources across a sufficiently large number of stocks from different sectors, investors completely eliminate portfolio risk, including the risk associated with recessions and changes in interest rates."( ) True ( ) False8) (Original question - CEBRASPE style, judge the statement) Judge the following statement: "When the correlation between the returns of two assets equals +1, diversification between them does not reduce risk, and the portfolio's standard deviation equals the weighted average of the individual standard deviations."( ) True ( ) FalseAnswer Key1) A — E[R] = 0.20 × (−5) + 0.50 × 10 + 0.30 × 20 = −1 + 5 + 6 = 10%.2) A — E[R] = 0.5 × 4 + 0.5 × 16 = 10%. Deviations: ±6 p.p.; variance = 0.5 × 36 + 0.5 × 36 = 36; standard deviation = √36 = 6%. Option B confuses variance (36) with standard deviation — the standard trick in this topic.3) A — ρ ranges from −1 (a perfect mirror image) to +1 (perfect synchronization), passing through 0 (independence). This range is what makes correlation diversification's secret ingredient.4) A — The lower the correlation, the greater the benefit; at the limit ρ = −1, the movements cancel each other out and risk can be reduced to zero with the right weights (the ice cream parlor × umbrellas pair).5) A — Systematic risk affects everyone (interest rates, recession, exchange rates, pandemic); specific risk affects one (fraud, strike, recall). The other options mix the two layers or present two risks of the same type.6) A — The lesson's central rule: with ρ < 1, portfolio risk falls below the weighted average of the risks (here, below 20%). It would be exactly 20% only with ρ = +1; zero would require ρ = −1 with appropriate weights.7) False — Diversification eliminates specific risk, but systematic risk (recession, interest rates) affects all stocks and remains — it is the irreducible floor of equities.8) True — With ρ = +1 there is no offsetting at all: the portfolio moves exactly like the average of the assets. This is the only case in which the naive intuition ("portfolio risk = average risk") is correct.
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